Andrew Phair1, Michael Brideson1, Jin Jin2,3,4, Mingyan Li2, Stuart Crozier2, and Lawrence Forbes1
1School of Natural Sciences, University of Tasmania, Hobart, Australia, 2School of Information Technology and Electrical Engineering, University of Queensland, Brisbane, Australia, 3ARC Training Centre for Innovation in Biomedial Imaging Technology, University of Queensland, Brisbane, Australia, 4Mark and Mary Stevens Neuroimaging and Informatics Institute, University of Southern California, Los Angeles, CA, United States
Synopsis
We
present WARF, a novel reconstruction algorithm for radial MRI data acquired
with a rotating radio-frequency coil (RRFC). The algorithm reconstructs each
pixel as a weighted sum of all acquired data, with the weights determined by
the k-space sampling pattern. The theory behind WARF leading to the derivation
of appropriate weights is presented, and then WARF is applied to both simulated
and experimental data sets. The results indicate WARF is achieving an improved
robustness to RRFC angular velocity variability and k-space trajectory
deviation compared with existing reconstruction methods.
Introduction
The rotating radio-frequency coil (RRFC) for MRI is an alternative to a
multi-coil array that achieves sensitivity over the entire subject region
through mechanical rotation during data acquisition whilst avoiding issues such as
coil-size restrictions and fabrication complexity1-6. Recently, the
RRFC has been combined with a radial k-space trajectory to allow
self-calibration through oversampled k-space centres7. Ideally, the
angular velocity and repetition time would be chosen to ensure that acquisitions
occur at several discrete and equidistant coil positions, allowing the data to
be treated as parallel array data. However, pneumatic actuation of the rotation
results in variation in the angular velocity, and acquisition positions become
dispersed. Jin et al.7 thus segmented spokes according to coil
position, allowing segments of data to be treated as ‘pseudo-coils’ for
parallel MRI reconstruction. Another approach, dynamic sensitivity averaging8,
relies on the assumption that a relatively uniform net sensitivity profile is
formed by the averaging of individual profiles, and reasons that by gridding
with a large-enough kernel a net sensitivity-weighted image can be obtained. Both
methods are sensitive to variation in coil velocity, and to deviation in
k-space sample locations, caused by rapidly switching gradient coils. Herein,
we present a weighted-sum approach to radial MRI image reconstruction with an
RRFC (WARF), which produces good quality images while providing improved
robustness to angular velocity variation and deviations in the k-space
trajectory.Method
We propose to
calculate each of the $$$N$$$ pixels in the reconstruction, $$$\tilde{\rho}_n$$$,
as a weighted sum of radial k-space samples $$$b_r$$$,
$$\tilde{\rho}_n=\sum_{r=1}^Rw_{rn}b_r,$$
where $$$w_{rn}$$$ are
weights that must be solved for. Each sample is allocated to one of $$$N_{c}$$$
pseudo-coils according to the coil position during acquisition, with $$$R_m$$$
points in the $$$m$$$th pseudo-coil. Utilising a discrete approximation, we write
each k-space sample as the Fourier transform of the sensitivity-weighted image,
$$b_r=\sum_{t=1}^{N}e_{rt}S^{(m_r)}_t\rho_t.$$
Here, $$$e_{rt}$$$ is
a Fourier coefficient and $$$S^{(m_r)}_t$$$ is the sensitivity of the $$$m_r$$$th
pseudo-coil at the $$$t$$$th pixel. Supposing the weights can be chosen such
that $$L_{nt}=\sum_rw_{nr}e_{rt}=\begin{cases}1,&t=n\\0,&t\neq
n\end{cases}$$
for every pseudo-coil,
then the first two equations yield $$\tilde{\rho}_n=\sum_{t=1}^NL_{nt}U_t\rho_t=U_n\rho_n,$$
where $$$U_n$$$ is the
net sensitivity at the nth pixel. Thus, finding weights to satisfy the $$$L_{nt}$$$
equation will yield an appropriate, sensitivity-weighted solution to the
reconstruction problem. Applying a Tikhonov regularisation yields, for every
pseudo-coil and every pixel,
$$(E^*E+\lambda{I})\mathbf{w}_n=E^*\mathbf{L}_n,$$
where $$$E$$$ is an $$$N\times{R_m}$$$
matrix of Fourier coefficients, $$$\lambda$$$ is a regularisation parameter, $$$\mathbf{L}_n$$$
is zero vector with a 1 in the $$$n$$$th position and $$$\mathbf{w}_n$$$ is a
vector containing weights for the $$$n$$$th pixel. This is a straightforward,
albeit computationally expensive, equation to solve $$$N_{c}\times{N}$$$ times.
However, as the weights depend only on the k-space sample locations, they can
be stored and need not be recalculated for every scan.
WARF was implemented in MATLAB (MathWorks, USA) and applied to RRFC data
simulated on an $$$804\times{134}$$$ radial trajectory, utilising a simulated sensitivity
profile (Figure 1) produced with FEKO (Altair, USA) for a coil with diameter
40mm, length 25.5mm and open angle 60°. Using a homogenous phantom and a mouse head
image, seventeen sets of simulations were performed; one with ideal parameters,
eight using variable velocity profiles, and eight with deviated k-space
trajectories.
The method was also applied to experimental scans of a shrew and a mouse
acquired on a 9.4T Bruker BioSpec 94/30 pre-clinical MRI Scanner with a UTE
sequence.
For comparison, four pre-existing reconstruction methods were applied to
each data set.Results
Figures 2 and 3 display the reconstructions for the ideal simulation,
and one example each of a variable velocity and deviated trajectory simulation
(corresponding to the velocity profile and trajectory in Figure 1). Figure 3
also displays the difference between the deviated trajectory reconstructions
and the original image, clearly demonstrating WARF artefacts are of a lower
magnitude than those of other methods. Mean square error (MSE) and structural
similarity (SSIM), as compared to the sensitivity-weighted original image, are
plotted for all seventeen simulations in Figure 4. These show WARF often
outperforms the other methods and is particularly apt at supressing artefacts
that result from trajectory deviations, achieving the lowest MSE value in every
such simulation considered, along with the highest SSIM value in 11 of the 16
simulations. The reconstructions from the experimental scans are presented in
Figure 5, where WARF reconstructions appear to have less artefacts than those
from GRAPPA, SENSE and SENSE+L1, while being of similar quality to dynamic
sensitivity averaging reconstructions.Discussion
Reconstruction quality is affected by the parameter $$$\lambda$$$, with
larger values resulting in reduced artefacts at the cost of additional
smoothing. Here, a value of $$$\lambda=0.3$$$ has been used, but this can be
adjusted depending on the relative importance of artefact suppression and image
sharpness.
Although WARF was designed for the radial RRFC imaging scheme described,
this is not explicitly introduced into the underlying theory. Thus, future work
will consider extending WARF to rotating arrays, stationary parallel MRI and
arbitrary k-space trajectories.Conclusion
We have introduced WARF, a new reconstruction algorithm for radial MRI
with an RRFC, and demonstrated it for both simulated and experimental data
sets, where it was seen to provide an improved robustness to variation in coil
velocity and deviations in k-space sample locations.Acknowledgements
This work was supported by the Australian Research Council Linkage
Projects (LP120200375), an Australian Government Research Training Program
(RTP) Scholarship, and was carried out using the Tasmanian Partnership for
Advanced Computing high-performance computing clusters.
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